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Sensitivity

Sensitivity. Joan Wrobel. Outline. What is Sensitivity & Why Should You Care? What Are Measures of Antenna Performance? What is the Sensitivity of an Interferometer? What is the Sensitivity of a Synthesis Image? Summary. What is Sensitivity & Why Should You Care?.

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Sensitivity

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  1. Sensitivity Joan Wrobel

  2. Outline • What is Sensitivity & Why Should You Care? • What Are Measures of Antenna Performance? • What is the Sensitivity of an Interferometer? • What is the Sensitivity of a Synthesis Image? • Summary Ninth Synthesis Imaging Summer School, Socorro, June 15-22, 2004

  3. What is Sensitivity & Why Should You Care? • Measure of weakest detectable emission • Important throughout research program • Sound observing proposal • Sensible error analysis in journal • Expressed in units involving Janskys • Unit for interferometer is Jansky (Jy) • Unit for synthesis image is Jy beam-1 • 1 Jy = 10-26 W m-2 Hz-1 = 10-23 erg s-1 cm-2 Hz-1 • Common current units: milliJy, microJy • Common future units: nanoJy Ninth Synthesis Imaging Summer School, Socorro, June 15-22, 2004

  4. Measures of Antenna Performance Source & System Temperatures • What is received power P ? • Write P as equivalent temperature T of matched termination at receiver input • Rayleigh-Jeans limit to Planck law P = kB´T´Dn • Boltzmann constant kB • Observing bandwidth Dn • Amplify P by g2 where g is voltage gain • Separate powers from source, system noise • Source antenna temperature TaÞ source power Pa = g2 ´kB´Ta ´Dn • System temperature Tsys Þ noise power PN = g2 ´kB´Tsys´Dn Ninth Synthesis Imaging Summer School, Socorro, June 15-22, 2004

  5. Measures of Antenna Performance Gain • Source power Pa = g2 ´kB´Ta´Dn • Let Ta = K´S for source flux density S, constant K • Then Pa = g2 ´kB´K´S´Dn (1) • But source power also Pa = ½ ´g2 ´ ha´A´S´Dn(2) • Antenna area A, efficiency ha • Rx accepts 1/2 radiation from unpolarized source • Equate (1), (2) and solve for K K = (ha´A) / (2 ´kB) = Ta / S • K is antenna’s gain or “sensitivity”, unit degree Jy-1 • K measures antenna performance but no Tsys Ninth Synthesis Imaging Summer School, Socorro, June 15-22, 2004

  6. Measures of Antenna Performance System Equivalent Flux Density • Antenna temperature Ta = K´S • Source power Pa = g2 ´kB´K´S´Dn • Express system temperature analogously • Let Tsys = K´SEFD • SEFD is system equivalent flux density, unit Jy • System noise power PN = g2 ´kB´K´SEFD´Dn • SEFD measures overall antenna performance SEFD = Tsys / K • Depends on Tsys and K = (ha´A) / (2 ´kB) • Examples in Table 9-1 Ninth Synthesis Imaging Summer School, Socorro, June 15-22, 2004

  7. Interferometer Sensitivity Real Correlator - 1 • Simple correlator with single real output that is product of voltages from antennas j,i • SEFDi = Tsysi / Ki and SEFDj = Tsysj / Kj • Each antenna collects bandwidth Dn • Interferometer built from these antennas has • Accumulation time tacc, system efficiency hs • Source, system noise powers imply sensitivity DSij • Weak source limit • S << SEFDi • S << SEFDj Ninth Synthesis Imaging Summer School, Socorro, June 15-22, 2004

  8. Interferometer Sensitivity Real Correlator - 2 • For SEFDi = SEFDj = SEFD drop subscripts • Units Jy • Interferometer system efficiency hs • Accounts for electronics, digital losses • E.g.: VLA continuum • Digitize in 3 levels, collect data 96.2% of time • Effective hs = 0.81 ´Ö0.962 = 0.79 Ninth Synthesis Imaging Summer School, Socorro, June 15-22, 2004

  9. Abscissa spans 30 minutes. Ordinate spans +/-300 milliJy. Interferometer Sensitivity Complex Correlator • Delivers two channels • Real SR , sensitivity DS • Imaginary SI , sensitivity DS • Eg: VLBA continuum • Figure 9-1 at 8.4 GHz • Observed scatter SR(t), SI(t) • Predicted DS = 69 milliJy • Resembles observed scatter Ninth Synthesis Imaging Summer School, Socorro, June 15-22, 2004

  10. Interferometer Sensitivity Measured Amplitude • Measured visibility amplitude Sm = • Standard deviation (s.d.) of SRor SIis DS • True visibility amplitude S • Probability Pr(Sm/DS) • Figure 9-2 • Behavior with true S/DS • High: Gaussian, s.d. DS • Zero: Rayleigh, s.d. DS x • Low: Rice. Smgives biased estimate of S. Use unbias method. Ninth Synthesis Imaging Summer School, Socorro, June 15-22, 2004

  11. Interferometer Sensitivity Measured Phase • Measured visibility phase fm = arctan(SI/SR) • True visibility phase f • Probability Pr(f-fm) • Figure 9-2 • Behavior with true S/DS • High: Gaussian • Zero: Uniform • Seek weak detection in phase, not in amplitude Ninth Synthesis Imaging Summer School, Socorro, June 15-22, 2004

  12. Image Sensitivity Single Polarization • Simplest weighting case where visibility samples • Have same interferometer sensitivities • Have same signal-to-noise ratios w • Combined with natural weight (W=1), no taper (T=1) • Image sensitivity is s.d. of mean of L samples, each with s.d. DS, i.e., DIm = DS/ÖL • N antennas, # of interferometers ½ ´N´ (N-1) • # of accumulation times tint/tacc • L = ½ ´N´ (N-1) ´ (tint/tacc) • So Ninth Synthesis Imaging Summer School, Socorro, June 15-22, 2004

  13. Image Sensitivity Dual Polarizations - 1 • Single-polarization image sensitivity DIm • Dual-polarization data Þ image Stokes I,Q,U,V • Gaussian noise in each image • Mean zero, s.d. DI = DQ = DU = DV = DIm/Ö2 • Linearly polarized flux density P = • Rayleigh noise, s.d. DQ ´ = DU ´ • Cf. visibility amplitude, Figure 9-2 • Polarization position angle c = ½ ´ arctan(U/Q) • Uniform noise between ± p/2 • Cf. visibility phase, Figure 9-2, ± p/2 Ninth Synthesis Imaging Summer School, Socorro, June 15-22, 2004

  14. NGC5548 FOV 150 milliarcsec = 80 pc Wrobel 2000, ApJ, 531, 716 Image Sensitivity Dual Polarizations – 2 • Eg: VLBA continuum • Figure 9-3 at 8.4 GHz • Observed • Stokes I, simplest weighting • Gaussian noise DI = 90 microJy beam-1 • Predicted DI = DIm/Ö2 = DS/ L = ½ ´N´ (N-1) ´ (tint/tacc) • Previous e.g. DS • Plus here L = 77,200 • So s.d. DI = 88 microJy beam-1 Ninth Synthesis Imaging Summer School, Socorro, June 15-22, 2004

  15. NGC5548 FOV 20 milliarcsec = 10 pc Wrobel 2000, ApJ, 531, 716 Image Sensitivity Dual Polarizations – 3 • Eg: VLBA continuum • Figure 9-3 at 8.4 GHz • Observed • Ipeak = 2 milliJy beam-1 • Gaussian noise DI = 90 microJy beam-1 • Position error from sensitivity? • Gaussian beam qHPBW = 1.5 milliarcsec • Then Dq = 34 microarcsec • Other position errors dominate Ninth Synthesis Imaging Summer School, Socorro, June 15-22, 2004

  16. Image Sensitivity Dual Polarizations – 4 • Eg: VLA continuum • Figure 9-4 at 1.4 GHz • Observed • Stokes Q, U images, simplest weighting • Gaussian DQ = DU = 17 microJy beam-1 • Predicted DQ = DU = DIm/Ö2 = DS/ L = ½ ´N´ (N-1) ´ (tint/tacc) • So s.d. DQ = DU= 16 microJy beam-1 Ninth Synthesis Imaging Summer School, Socorro, June 15-22, 2004

  17. Mrk231 FOV 70 arcsec = 60 kpc Ulvestad et al. 1999, ApJ, 516, 127 Image Sensitivity Dual Polarizations – 5 • Eg: VLA continuum • Figure 9-4 at 1.4 GHz • Observed • Stokes I image • Simplest weighting • Gaussian noise ΔI > ΔQ = ΔU • Expect s.d. DI = DQ = DU = DIm/Ö2 if each image limited by sensitivity • Other factors can increase DI • Suspect dynamic range as Ipeak = 10,000 DI • Lesson: Use sensitivity as tool to diagnose problems Ninth Synthesis Imaging Summer School, Socorro, June 15-22, 2004

  18. Sensitivity Summary – 1 • One antenna • System temperature Tsys • Gain K • Overall antenna performance is measured by system equivalent flux density SEFD SEFD = Tsys / K • Units Jy Ninth Synthesis Imaging Summer School, Socorro, June 15-22, 2004

  19. Sensitivity Summary - 2 • Connect two antennas to form interferometer • Antennas have same SEFD, observing bandwidth Δν • Interferometer system efficiency hs • Interferometer accumulation time tacc • Sensitivity of interferometer • Units Jy Ninth Synthesis Imaging Summer School, Socorro, June 15-22, 2004

  20. Sensitivity Summary - 3 • Connect N antennas to form array • Antennas have same SEFD, observing bandwidth Dn • Array has system efficiency hs • Array integrates for time tint • Form synthesis image of single polarization • Sensitivity of synthesis image • Units Jy beam-1 Ninth Synthesis Imaging Summer School, Socorro, June 15-22, 2004

  21. Westerbork’s Most Sensitive Image A Deep WSRT 1.4 GHz Radio Survey of the Spitzer Space Telescope FLSv Region, Morganti et al. 2004, A&A, in press, astro-ph/0405418 ΔI = 8.5 microJy beam-1 Ninth Synthesis Imaging Summer School, Socorro, June 15-22, 2004

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